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Counting elliptic curves with prescribed torsion

2015; De Gruyter; Volume: 2017; Issue: 729 Linguagem: Inglês

10.1515/crelle-2014-0107

1435-5345

Robert Harron, Andrew Snowden,

Cryptography and Residue Arithmetic

Abstract Mazur’s theorem states that there are exactly fifteen possibilities for the torsion subgroup of an elliptic curve over the rational numbers. We determine how often each of these groups actually occurs. Precisely, if G is one of these fifteen groups, we show that the number of elliptic curves up to height X whose torsion subgroup is isomorphic to G is on the order of X 1 / d {X^{1/d}} , for some number d = = d ( G ) which we compute.